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Other literature type . 2026
License: CC BY
Data sources: Datacite
ZENODO
Other literature type . 2026
License: CC BY
Data sources: Datacite
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ONE AXIOM : The Critical Ridge : Phase Coherence as a Structural Invariant of ζ(s)

The Critical Ridge
Authors: Spychalski, Robert;

ONE AXIOM : The Critical Ridge : Phase Coherence as a Structural Invariant of ζ(s)

Abstract

7B — The Critical Ridge: Phase Coherence as a Structural Invariant of $\zeta(s)$ Description: Overview This paper presents a formal structural establishment of the Geometric Riemann Hypothesis (G-RH) within the ONE AXIOM framework. Moving beyond traditional analytic number theory, document 7B identifies the distribution of Riemann zeros as a structural necessity of holographic decompression. By employing a Dual-Track Methodology, the work bridges Lens 2 (Geometric/Classical) and Lens 4 (Ontological), revealing RH as a compatibility condition between the VOID ($\pi_4$) and the relational layer ($\pi_6$). Key Contributions The Bridge Theorem: The central result of the paper, establishing that G-RH implies the Classical Riemann Hypothesis. The theorem demonstrates that phase pinning at $\Phi \in \{0, \pm\pi\}$ and the exclusion of off-axis quadruplets are mandatory for ontological coherence. Possibility Selection Principle (PSP): Application of the 51:13 structural filter derived from Fano-K7 duality. The paper demonstrates that 13 out of 64 phase sectors are "FORBIDDEN," preventing any off-axis zero formation. Statistical Validation: A rigorous analysis of the first 400 Riemann zeros shows an overwhelming fit for the ONE AXIOM model ($\chi^2 = 713.22$) with a Bayes Factor ($BF > 10^{154}$), ruling out the null hypothesis of random distribution. Independent Experimental Validation: Integration of recent findings by Wei et al. (arXiv:2511.11199), providing external empirical support for the structural predictions of the model. Quantum Gap Analysis: The analysis reveals a 90-point quantum gap: 100% of on-critical zeros occupy ALLOWED sectors, versus only 10.2% of off-critical points. This disparity confirms the "Critical Ridge" as a preferred state of ontological stability. Methodology & Formalism The work introduces an "Epistemological Switch," shifting from numerical verification to structural recognition. It contains 8 formal theorems (Section 8) that define the Ontological Coherence Field (OCF) and the mechanism of phase-locking. The paper formalizes the condition $\sigma = 1/2 \iff \Phi \in \{0, \pm\pi\}$ on the critical line, linking Hardy’s Z-function to the $q^ = 3/2$* Tsallis critical exponent at Layer $\pi_{5.5}$. Target Audience Researchers in complex analysis, quantum chaos, and theoretical physics interested in non-extensive statistics, holographic principles, and the structural foundations of mathematics.

Keywords

Possibility Selection Principle (PSP), Wei et al. Validation (arXiv:2511.11199), Critical Ridge ($\pi_{5.5}$), Bridge Theorem, Riemann Hypothesis, Tsallis Statistics ($q^*$), ONE AXIOM Framework, ONE AXIOM, Phase Coherence ($\Phi$)

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
0
Average
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